Neumann-Neumann methods for a DG discretization on geometrically nonconforming substructures - Dryja, Maksymilian; Galvis, Juan; Sarkis, Marcus (Numerical methods for partial differential equations, 2012)
A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite element space associated to a triangulation \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} {\mathcal{T}}_{h_i}(Ω_i)\end{align*}\end{document} is introduced. To handle the nonmatching meshes across ∂Ωi, a discontinuous Galerkin discretization is considered. In this article, additive and hybrid Neumann‐Neumann Schwarz methods are designed and analyzed. Under natural assumptions on the coefficients and on the mesh sizes across ∂Ωi, a condition number estimate \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*} C(1 + \max_i\log \frac{H_i}{h_i})^2\end{align*}\end{document} is established with C independent of hi, Hi, hi/hj, and the jumps of the coefficients. The method is well suited for parallel computations and can be straightforwardly extended to three dimensional problems. Numerical results are included. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012